Some very common, often unrecognized, problems
OFOT: When two factors are essential
(like the length and width of a corn field for calculating its area) it's
oversimplification to declare one false statement and one true statement
(OFOT) like this: "It not the width that determines the area; it's the length!"
"Guns don't kill people; people kill people." Sure, OFOT is absurd;
but it's heard over and over again in daily discussions. Furthermore,
it's about the simplest case of a much larger family of errors of omission
in situations of multiple influences—which is just about every situation.
Necessity vs. Sufficiency: Rejecting
a necessity on the grounds that it isn't sufficient is oversimplification.
("That no panacea; ignore it!" "Science cannot answer questions
about human values, so science is just an alternative reality which I don't
choose to accept." ) Claiming sufficiency on the basis of necessity is
oversimplification. ("We are seeking the simple differential
equations that describe, predict, and explain everything.") These
are two kinds of improperly inverted implications.
Implication, the logical relationship:
(Necessity vs sufficiency is a special case.) If A implies
is sufficient for B; but the inverse relationship,
means that A is necessary for
B. Notice that
when expressed in this abstract way, the relationships are at least a little
confusing to a large fraction of people—but are
"perfectly obvious" to a small fraction of people. On
the other hand, specific, familiar examples are quite "obvious" to a majority.
For example, "If there is a potato in the box, then there is a vegetable
in the box" (potato is sufficient for being a vegetable) is pretty obvious
to most. Furthermore, "If there is a vegetable in the box, there
may or may not be a potato in the box" (a vegetable is not
necessarily a potato) is equally obvious. For those relatively few
who "see" the
abstract expression, a specific kind of "perception
of abstraction" exists. Those individuals have developed a sense
for implication which turns out to be a powerful tool for "seeing" certain
science concepts and even for extending them. Without
that "sixth sense," a person may see implication, and perhaps even science
itself, as "merely a social construct with no universal principles..."
Familiar statements that indicate misunderstanding. "The
Singles" is a persistent, pervasive, and pernicious infection of human
use of knowledge. "Perceptions of abstractions" that organize our thoughts
about multi-element relationships can help vaccinate us against "The Singles."
kind of "perception" is remarkably resistant to teaching and learning,
and that fact contributes strongly to the perceived difficulty of science
and mathematics. On our Web site we refer to this problem as "exploring
the edges of human comprehension." Our raw material for
examining this problem and for constructing a useful theory of the edges
of human comprehension is a collection of "Statements that give little
clues that something wasn't really quite understood." This collection
is at our Misunderstandings Workshop.
We invite everyone to contribute to this collection and to the interpretations
suggested. If you have a Web site that discusses such matters we
would like to incorporate links to your site.
Rank-orderable measure. Comparatives
and superlatives require scalar (single-component) measure. Most
measures have multiple components. Comparatives and superlatives
rarely have the meaning intended. More often they express personal
biases. Overcoming scalar-limited
perception of measurement is a challenge at the edge of human comprehension
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